Download - Poisson limit
5 10 15 200
1
Poisson limit
HTHT
HT
HT
HT
HT
Hπββ β¨π₯ β©=π π
π=β¨π₯ β©πβ0
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©π (π₯ )= ππ₯
π₯ !πβπ
Distribution plot adapted from (http://en.wikipedia.org/wiki/File:Poisson_pmf.svg), licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.
π₯T T T T
π=1
π=4π=10
T
π (π₯ )
0.1
0.2
0.3
0.4
HTHT
HT
HT
HT
HT
H
2
Poisson limit
?π=1π=1
β¨π₯ β©=1
π=2π=1 /2
β¨π₯ β©=1 ? ?
? ? ? ?π=4π=1 /4
β¨π₯ β©=1
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
Average total number of heads
T T
3
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
Variance of total number of heads
β¨ πΏπ₯2 β©=ππ (1βπ )β¨π₯ β©π
β¨π₯ β©π
β¨ πΏπ₯2 β©= β¨π₯ β©
T T
4
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©
Probability distribution for getting x total heads
π (π₯ )= π !(πβπ₯ ) !π₯ !
π π₯ (1βπ )πβπ₯
(ππ₯
π )π₯ (1β ππ₯
π )πβπ₯
(1β ππ₯
π )π (1β ππ₯
π )βπ₯
β1πβππ₯
π (π₯ )= π !(πβπ₯ ) !π₯ ! (ππ₯
π )π₯
πβππ₯
T T
5
Poisson limit
HTHT
HT
HT
HT
HT
H
πβββ¨π₯ β©=π π
π=β¨π₯ β©πβ0
H
T T T T T T. . .
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©
Probability distribution for getting x total heads
π (π₯ )= π !(πβπ₯ ) !π₯ ! (ππ₯
π )π₯
πβππ₯
π β (πβ1 ) β (πβ2 )β― (πβπ₯+1 ) β (πβπ₯ ) !(πβπ₯ )!π₯ !
π (π₯ )=π β (πβ1 ) β (πβ2 )β― (πβπ₯+1 )π₯ ! (ππ₯
π )π₯
πβππ₯
π (π₯ )=ππ₯
π₯
π₯ !πβππ₯
T T T
π (π₯ )
0.1
0.2
0.3
0.4
5 10 15 200
6
HTHT
HT
HT
HT
HT
Hπββ β¨π₯ β©=π π
π=β¨π₯ β©πβ0
. . .
β¨ πΏπ₯2 β©= β¨π₯ β©π (π₯ )= ππ₯
π₯ !πβπ
Distribution plot adapted from (http://en.wikipedia.org/wiki/File:Poisson_pmf.svg), licensed by Wikipedia user Skbkekas under a CC-BY-3.0 license.
π₯T T
π=1
π=4π=10
Poisson limit